Notes for Math 663 Spring 2016demlow/Courses/16_sp/663_sp16/...Notes for Math 663 Spring 2016 Alan...

Notes for Math 663Spring 2016

Alan Demlow

Chapter 1

Introduction and preliminaries

1.1 Introduction

In this chapter we will introduce the main topics of the course. We first briefly define the finiteelement method and state a basic error estimate.

Let ⌦ ⇢ Rd (d = 2, 3) be a polygonal or polyhedral domain, and let f 2 L2(⌦). Consider theboundary value problem

��u = f in ⌦,

u = 0 on @⌦.

Rewriting in weak form, we seek u 2 H10 (⌦) such that

a(u, v) :=

Z

⌦rurv dx =

Z

⌦fv dx =: (f, v), v 2 H1

0 (⌦).

Here we define

Hm(⌦) = {u : D↵u 2 L2(⌦), all multiindices ↵ with |↵| m}

andH1

0 (⌦) = {u 2 H1(⌦) : tru = 0 on @⌦}.

Here tr is the trace operator.In order to define a finite element method, let Th be a regular simplicial decomposition of ⌦.

That is, ⌦ = [T2ThT , and the intersection of any two members of Th is either empty or consists ofa common lower-dimensional simplicial boundary component of each member (i.e., a face, edge, orvertex if d = 3, or an edge or vertex if d = 2). We additionally assume that for each T 2 Th, thereare balls bT ⇢ T ⇢ BT such that diam bT ' diamBT ' diamT . Here we use the shorthand a ' bto mean that a c1b c2a with constants c1, c2 independent of essential quantities. For the timebeing, we also assume that Th is quasi-uniform, that is, diamT ' diamT 0 for all T, T 0 2 Th.

Next let Vh = {u 2 C(⌦)\H10 (⌦) : u|T 2 Pr} for some fixed r � 1. r = 1 corresponds to the case

of piecewise linear finite element spaces, r = 2 to piecewise quadratics, etc. We then seek uh 2 Vh

such thata(uh, vh) = (f, vh), vh 2 Vh.

3

4 CHAPTER 1. INTRODUCTION AND PRELIMINARIES

Due to the coercivity of the form a(·, ·) on H10 (⌦), we may apply Cea’s Lemma to find that

ku� uhkH10 (⌦) = inf

�2Vh

ku� �kH10 (⌦).

Application of a suitable interpolation operator then implies that

inf�2Vh

ku� �kH10 (⌦) Chr|u|Hr+1(⌦),

assuming that u 2 Hr+1(⌦). That is, the finite element error decreases with order hr:

ku� uhkH1(⌦) Chr|u|Hr+1(⌦).

From approximation theory we can conclude that this is the best possible order of convergence whenusing Lagrange elements. This error estimates thus tells us that in a certain (rather basic) sense,we are getting what we paid for out of our finite element method–assuming that u is su�cientlysmooth.

The question naturally arises, then, whether the assumption u 2 Hr+1(⌦) is realistic. If theboundary @⌦ is smooth, then a shift theorem holds:

kukHm+2(⌦) Ck�ukHm(⌦), m � �1, (1.1)

whenever ��u = f 2 Hm(⌦). When @⌦ is not smooth, such an estimate does not hold in general.We then may ask some natural questions: For what combinations of domains ⌦, parameters r,and data functions f can we expect that u 2 Hr+1(⌦)? What happens if u /2 Hr+1(⌦)? Can wesomehow fix the error estimate or our method to ensure that we can still get optimal-order errordecrease from our method in this situation?

We also recall a standard L2 error estimate and outline its proof. Assume now that (1.1) holdswith m = 0, and let z 2 H1

0 (⌦) solve

a(v, z) = (v, u� uh), v 2 H10 (⌦).

Then for any zh 2 Vh,

ku� uhk2L2(⌦) = a(u� uh, z) = a(u� uh, z � zh),

since a(u,�) = (f,�) = a(uh,�) and so a(u � uh,�) = 0, � 2 Vh (Galerkin orthogonality). Thenby using Cauchy-Schwarz, choosing zh so that kz� zhkH1(⌦) Ch|z|H2(⌦), and employing (1.1), wehave

ku� uhk2L2(⌦) ku� uhkH1(⌦)kz � zhkH1(⌦) Chku� uhk|z|H2(⌦) Chku� uhk2L2(⌦).

Thus assuming that u 2 Hr+1(⌦), we have

ku� uhkL2(⌦) Chku� uhkH1(⌦) Chr+1|u|Hr+1(⌦).

We note that (1.1) holds for m = 0 whenever ⌦ is convex, and in particular when ⌦ is convexand polygonal. However, it does not generally hold for m > 0 even if ⌦ is convex, though there aresome convex polygonal domains for which (1.1) does hold for m = 1, such as squares and rectangles.

1.1. INTRODUCTION 5

If ⌦ is a non-convex polygonal or polyhedral domain, then (1.1) never holds for m = 0. Thus theabove L2 error estimate can not be assumed to be valid in such cases as the above poof techniquedoes not apply, and indeed a reduction in order of convergence generally can be observed in suchcases.

In this course, we will study finite element error behavior in the presence of singularities, whichare points where the solution u or one of its derivatives fails to be defined. In such cases the rangeof values of r for which u 2 Hr+1(⌦) as required above is generally restricted. Our main focus willbe singularities caused by non-smoothness of the boundary @⌦, and in particular on polygonal andpolyhedral domains ⌦. As we noted above, convexity or nonconvexity of the domain already has apronounced e↵ect on the regularity of solutions to Poisson’s problems. The techniques we use willallow us to make finer distinctions on the smoothness of such solutions based on geometric propertiesof the domain. In addition, these regularity properties in turn a↵ect the provable convergence ratesof the FEM in the energy norm as well as other norms such as the L2 norm. We will delve furtherinto this issue as well.

The twin tasks of understanding error behavior and devising ways to improve numerical perfor-mance in the presence of such singularities have drawn attention from finite element researchers sinceat least the 1970’s. Early e↵orts in this direction tended to focus on a priori error estimation andmanual construction of graded (locally refined) meshes which could appropriately resolve singulari-ties having known properties. While helpful in understanding basic method and error behavior, thisapproach is practically limited. For linear problems, singularity locations can generally be predictedbased on problem data such as f and elliptic coe�cients. However, the singularity strength candepend on these data, and even for the Laplacian it is somewhat involved to meaningfully predictsingularity properties strengths especially in three space dimensions. For nonlinear problems, thesituation is even more complex as singularity location also cannot generally be predicted a priori.

For these reasons, self-adaptive algorithms are a popular tool for e�ciently resolving singularitiesand other significant local variations in the solution u. These algorithms automatically detectsingularities and direct e�cient local mesh refinement. Adaptive finite element methods (AFEM)have also been in existence for many years (at least since the 1980’s). Meaningful convergenceanalysis of such algorithms is however relatively recent, with a mature theory for basic ellipticproblems occurring only within the past decade. Convergence theory for adaptive finite elementalgorithms continues to be a very active area of research.

The main goal for the course, broadly speaking, is to understand the interplay between regularityof solutions to boundary value problems and convergence rates of finite element methods. Morespecifically, our goals are to:

1. Understand the basic structure of singularities arising in elliptic boundary value problems onpolygonal and polyhedral domains.

2. Learn various ways of measuring the smoothness (regularity) of solutions to PDE. Examplesmay include Wm

p(Sobolev) spaces, fractional-order Sobolev spaces, weighted Sobolev spaces,

Besov spaces, and Holder spaces.

3. Gain a brief understanding of a priori mesh grading techniques for resolving singularities, andthe e↵ects of systematic grading on a priori error estimates.

4. Understand various notions of optimality of finite element methods.


5. Gain an in-depth understanding of adaptive convergence theory and the ability of adaptivemethods to resolve singularities.

6. As time and interests of the audience allow, study finite element error behavior (a priori anda posteriori) in nonstandard norms, including local energy norms and the maximum norm.

There are a number of extensions of these topics that we may either cover briefly or whichwould make good student projects. These include convergence of adaptive algorithms for othertypes of elliptic problems and FEM (Stokes’, Maxwell’s equations, elliptic equations in mixed form),convergence of adaptive hp-FEM, singularities and adaptivity for eigenvalue problems, anisotropicmesh refinement, and others.

1.2 Computational prologue

In order to illustrate some of the topics mentioned above, we return to the finite element errorestimate

ku� uhkH1(⌦) Chr|u|Hr+1(⌦). (1.2)

when the finite element method uses polynomials of degree r. This estimate seems to imply that itis more e�cient to use higher polynomial degrees of finite element spaces. To test this assumption,we discuss what actually happens in computational practice.

Also, We first reorient our definition of order of convergence by expressing it with respect to thenumber of degrees of freedom in the finite element space instead of h. Assume that we have a quasi-uniform grid. Shape-regularity implies that each T 2 Th has volume equivalent to hd. Thus thenumber of elements in the grid is equivalent (up to some fixed constant) to volume(⌦)/hd ' h�d. LetDOF (“degrees of freedom”) be the size of the finite element system. We then have for quasi-uniformgrids that

DOF ' h�d, h ' DOF�1/d.

On a quasi-uniform grid, (1.2) then translates to

ku� uhkH1(⌦) . DOF�r/d|u|Hr+1(⌦). (1.3)

By a . b we mean a Cb for some nonessential constant C, and by a ' b we mean that a . band b . a. The advantage of expressing rates of convergence using numbers of degrees of freedominstead of h is that this notion applies and is meaningful even if the grid is not quasi-uniform. Thenumber of degrees of freedom is also a reasonable measure of the computational cost of the finiteelement algorithm, since with proper solver technology the overall cost of the algorithm increasesroughly linearly with DOF .

We take the right-hand-side data (forcing function) to be f(x) = 1 and solve ��u = f , u = 0 on@⌦ for a few di↵erent choices of ⌦. Our actual computations mostly assume r = 1 (piecewise linearelements), but we also discuss what happens for di↵erent choices of r. We don’t know u in thesecomputations, but we can still measure ku� uhkH1(⌦) by employing an a posteriori error estimator⌘ which satisfies ⌘ ' ku� uhkH1(⌦). (More precisely, we use a residual-type error estimator. As wewill learn later in the course, for general f we have equivalence between error and estimator only upto data oscillation, but for f ⌘ 1 data oscillation is zero and so the estimator is always equivalentto the error.)

1.2. COMPUTATIONAL PROLOGUE 7

102 103 104 105

Number of unknowns

10-1

100

Erro

r

Rate of convergence is CN-0.49631

2

CN-0.49631

Figure 1.1: Error reduction on a square domain.

First we investigate convergence on the square ⌦ = (�1, 1) ⇥ (�1, 1). This is a convex domain,so we have u 2 H2(⌦) and kukH2(⌦) . kfkL2(⌦) . 1. We then expect that for a uniform sequenceof mesh refinements,

ku� uhkH1(⌦) . h|u|H2(⌦) ' DOF�1/2,

since h ' DOF�r/d = DOF�1/2. Computations are performed using the MATLAB toolbox iFEMreference?, and rates of convergence are estimated using a log-least squares fit of the estimator. Theestimated rate of convergence (which is very close to the theoretical one of 0.5) is plotted in Figure3.1 and one of the computational meshes and the solution in Figure 1.2.

We next work on an “L-shaped domain”, realized here as ⌦ = (�1, 1)⇥ (�1, 1) \ [0, 1)⇥ (�1, 0].Because the domain is not convex, we don’t expect the H2 regularity result kukH2(⌦) . kfkL2(⌦)

to hold, and in fact it does not. At least our proof that ku � uhkH1(⌦) . DOF�1/2 does not holdhere. In Figure 1.3 we display the calculated rate of convergence on a quasi-uniform mesh. Therewe obtain roughly that ku � uhkH1(⌦) . DOF�.37, More careful numerical calculation (or theory)

would show a convergence rate of DOF�1/3 for su�ciently large DOF (note that the observedconvergence line is slightly more shallow than the reference line for high DOF in Figure 1.3). Thisis a worse convergence rate than we experienced when computing on the unit square, so in somesense we aren’t getting what we paid for out of our piecewise linear elements.

We finally perform the same computation on the “crack domain”, which is a square with aslit removed (in this case the square with vertices (1, 0), (0, 1), (�1, 0), and (0,�1), minus thenonnegative real axis). Figure 1.5 and ?? for the observed convergence rate and solution. Theobserved convergence rate is about DOF�.262, which is expected to settle down to DOF�.25 for


Figure 1.2: Mesh and solution on square domain.

102 103 104 105 106

Number of unknowns

10-1

100

Erro

r


2

CN-0.37145

Figure 1.3: Error reduction on an L-shaped domain.


Figure 1.4: Mesh and solution on an L-shaped domain.

su�ciently large DOF . Thus we see a reduction in convergence rate from DOF�.5 to DOF�.33

to DOF�.25 as we move from the square to the L-shaped to the crack domain. This indicates areduction in the regularity of the boundary value problem as we make the domain less convex, sinceall other problem parameters are the same here. We note also that while (1.3) indicates that wemight expect a better rate of convergence if we increase the polynomial degree, that will not happenat all on the L-shaped or crack domains if we use only quasi-uniform meshes (it will at least to anextent on the square domain).

Because quasi-uniform meshes don’t give us what we have paid for on non-convex domains evenfor piecewise linear elements, we now consider the use of solution-adapted meshes. That is, we willlocally refine (grade) the mesh in such a way that the solution is resolved more e�ciently. Herewe use an adaptive finite element method to accomplish this, but in these simple two-dimensionalcases essentially the same results can be obtained using manual (a-priori) mesh grading. We runour adaptive method on the unit square, L-shaped domain, and crack domain as above. On theunit square (Figure 1.6 and 1.7), we obtain optimal-order convergence rate as with uniform meshrefinement. There is moderate grading in the solution-adapted mesh, and perhaps a slight reductionin the error reached for a given number of degrees of freedom, but generating a solution-adaptedmesh is not worth the trouble in this particular case. (It might be for the same problem if onewished to use higher polynomial degrees.) On the L-shaped and crack domains, however, we onceagain achieve a rate of convergence of about DOF�.5. Thus using solution-adapted meshes lets usget what we paid for from our algorithm. The obtained meshes (Figures 1.9 and Figure 1.11) showthat mesh elements are concentrated near the reentrant (non-convex) corner of both of the domains.This occurs because strong singularities are generated there. They are the reason that u /2 H2(⌦)for these problems and the use of solution-adapted meshes is thus necessary to recover optimalconvergence rates. We note also that by using adaptivity, for these two-dimensional problems onecan always recover the energy-norm convergence rate DOF�r/2.

The situation in three space dimensions is somewhat di↵erent. In two dimensions, we indicatedabove that adaptivity generally allows us to recover the “optimal” convergence rate DOF�r/d =


101 102 103 104 105 106

Number of unknowns

10-1

100

Erro

r


2

CN-0.26293

Figure 1.5: Error reduction on a crack domain.

102 103 104

Number of unknowns

10-1

100

Erro

r


2

CN-0.50083

Figure 1.6: Adaptive error reduction on a square domain.


Figure 1.7: Adaptive mesh on a square domain.

102 103 104

Number of unknowns

10-1

100

Erro

r


2

CN-0.49222

Figure 1.8: Adaptive error reduction on an L-shaped domain.


Figure 1.9: Adaptive mesh on an L-shaped domain.

101 102 103 104 105

Number of unknowns

10-2

10-1

100

Erro

r


2

CN-0.48371

Figure 1.10: Adaptive error reduction on a crack domain.


12

3 4

1

2

3

4

5 16

Figure 1.11: Adaptive mesh on a crack domain.

DOF�r/2. To test whether this is also true in 3D, consider the domain ⌦ pictured in Figure 1.12.This is a convex polyhedral domain, so for the case r = 1 we expect optimal-order convergence (nowDOF�1/3). For r = 2 and r = 3, we can also recover optimal-order convergence rates DOF�2/3 andDOF�1 by using adaptivity. However, for r = 4 (quartic elements), even with adaptive meshes weonly recover a convergence rate of about DOF�8/7 (the line “H1refinement/H1 estimator”, havinglogarithmic slope �1.12). Using adaptivity for any r > 4 will lead to similar results. On non convexdomains a similar barrier in convergence rates will be reached for lower polynomial degrees. Insummary, we make the following observations:

1. In general for polygonal and polyhedral domains, he more highly non-convex the domain is,the worse the convergence rate on quasi-uniform meshes.

2. When solving Poisson’s problem on 2D polygonal domains with smooth f , we can alwaysrecover optimal convergence rates by using solution-adapted meshes.

3. Adaptivity helps in three space dimensions, but does not always allow us to recover optimalconvergence rates.

Much of the course will be spent explaining these observations. That is, we ask and answer thefollowing questions:

1. What causes increasing reduction in convergence rates on quasi-uniform meshes for increasinglynon-convex domains? (We answer this question by explaining the singular structure of solutionson polyhedral domains.)


Figure 1.12: 3D convex polyhedral domain.

2. How do we adapt meshes to e�ciently resolve singular solutions? (We will give two di↵erentanswers to this question, one involving a priori “by hand” mesh gradings and the other involvingautomatic adaptivity.)

3. What’s the di↵erence between two and three space dimensions? Why can we generally recoveroptimal convergence rates in 2D by using adaptivity, but not always in 3D? (The answerrequires a good understanding of the involved singularities in each case, and the notion ofanisotropic functions which which have di↵erent regularity in di↵erent directions.)

1.3 Analytical preliminaries

1.3.1 Domain properties

Classification of domains plays an important role in analyzing boundary value problems and nu-merical methods for solving them. By domain we mean a nonempty open set in Rd. We generallyassume that ⌦ is in addition bounded.

We first define Ck domains. We say that the boundary @⌦ of ⌦ is Ck if for each x0 2 @⌦ thereexists r > 0 and a Ck function � : Rd�1 ! Rd such that (upon reorienting the coordinate axes ifnecessary)

⌦ \Br(x0) = {x 2 Bx0(r) : xd > �(x1, ..., xd�1)}. (1.4)

We generally denote by ⌫ or ~n the outward-pointing unit normal vector to @⌦.

1.3. ANALYTICAL PRELIMINARIES 15

-8.5!

-7.5!

-6.5!

-5.5!

-4.5!

-3.5!

-2.5!

2! 3! 4! 5! 6! 7!

Slope=-1.01!L2 refinement/H1 estimator!H1 refinement/H1 estimator!Slope=-1.12!H1 refinement/L2 estimator!Slope=-1.47!L2 refinement/L2 estimator!slope=-1.65!

Figure 1.13: Adaptive convergence rates on 3D convex polyhedral domain.


Let � : ⌦ ! R. � is Lipschitz if |�(x)� �(y)| L|x� y| for all x, y 2 ⌦. L is then the Lipschitzconstant of �. (Note that all C1 functions are Lipschitz, but not vice-versa.) � is Holder continuouswith exponent ↵ (0 < ↵ < 1) if |�(x) � �(y)| C↵|x � y|↵, x, y 2 ⌦. A domain ⌦ has Lipschitzboundary if in the definition of Ck domain we replace the requirement that the functions � be Ck

with the requirement that they be Lipschitz. We can also extend the definition of Ck boundaries toCk,↵ boundaries by requiring that the functions � in the above definition be k times di↵erentiablewith all k-th derivatives Holder continuous with exponent ↵.

We say that a domain ⌦ ⇢ R2 is polygonal if its boundary can be broken into a set of edgesconsisting of open line segments and vertices. A three dimensional domain ⌦ ⇢ R3 is polyhedral ifits boundary can be decomposed into open polygonal faces, open edges, and vertices. The definitionof polyhedral (polygonal) domains can be generalized to the notion of domains having polyhedral(polygonal) structure. There the definition of edges and faces above is generalized to include smoothcurve segments in the case of edges and the images of polygons under smooth mappings in the caseof faces. While we only consider polyhedral domains below, domains with polyhedral structure areimportant as they describe objects such as aircraft wings that have both curved smooth portionsand edges (and possibly vertices). Much of what we will learn about boundary value problems onpolyhedral domains applies as well to domains with polyhedral structure.

Finally, we say that ⌦ satisfies the cone condition if there exists a finite cone C such that eachx 2 ⌦ is the vertex of a finite cone Cx contained in ⌦ and congruent to C.

It is important to understand the relationship between various notions of smoothness of domain.All Ck domains (k � 1) are Lipschitz. There is not an inclusion in either direction between thesets of Lipschitz and polyhedral domains. That is, not all polyhedral domains are Lipschitz, andvice-vera. All polyhedral domains do however satisfy the cone condition. Examples of domains notsatisfying the cone condition include those having cusps on the boundary.

Over the past several decades much e↵ort has been devoted to the study of elliptic boundaryvalue problems on domains with nonsmooth boundaries. There is a large body of literature devotedto the study of BVP’s on Lipschitz domains, and there are many analytical results that are usefulto numerical analysts that have been proved under the assumption that ⌦ is Lipschitz. Such resultscan also often be useful when considering boundary value problems on polyhedral domains sincemost (but not all) interesting polyhedral domains are Lipschitz. There are however importantcounterexamples. These include domains with cracks, as in the computational example in theprevious subsection. Another well-known counterexample is the “two-brick” domain (Figure 1.14),which fails to be Lipschitz because the normals on faces near the origin (denoted by the arrow inthe illustration) point in opposite directions. One might think that boundary value problems onnon-Lipschitz domains might behave worse than on Lipschitz domains, but in the case of polyhedraldomains that is not really the case. Polyhedral structure is special and worth studying in its own rightin the context of boundary value problems, and the non-Lipschitz nature of these counterexamplesdoes little to change the properties of Lipschitz boundary value problems posed on them.

As a final note, there are also more general classes of non smooth domains that have appeared inthe analysis literature and which may be useful for analyzing finite element methods. For example,domain decomposition algorithms can lead to consideration of “jagged” subdomains consisting ofsomewhat arbitrary unions of mesh elements. Such domains might not have uniformly Lipschitzcharacter, so they have sometimes instead been characterized as John domains.


Figure 1.14: Non-Lipschitz “two-brick” domain.


1.3.2 Function spaces

Here we discuss Sobolev spaces (mainly); cf. the well-known books by Adams and Fournier [1, 2]for more details.

Given a domain ⌦, Lp(⌦) is the space of p-integrable measurable functions on ⌦ (1 p < 1)and L1(⌦) the space of essentially bounded functions. A function u 2 L1(⌦) has a weak derivativeD↵u (with ↵ a multi index) if for all � 2 C1

0 (⌦),

Z

⌦uD↵� dx = (�1)|↵|

Z

⌦�D↵u dx.

We then set for 1 p < 1

W p,k(⌦) = {u 2 Lp(⌦) : D↵u 2 Lp(⌦), |↵| k},

kukWp,k(⌦) = (X

|↵|k

kD↵ukpLp(⌦))

1/p. (1.5)

For p = 1, we have instead kvkWk1(⌦) = max|↵|k kD↵vkL1(⌦). Note that W p,k is a Banach space

(complete linear space), and that when p = 2 it is a Hilbert space.When 1 p < 1, Ck(⌦) (the k-times continuously di↵erentiable functions on ⌦ is dense in

W p,k(⌦). That is, every u 2 W p,k(⌦) can be approximated by a sequence un 2 Ck(⌦) such thatku � unkWp,k(⌦) ! 0 as n ! 1. This is true for any domain ⌦ (cf. Theorem 3.17 of [2]). Sucha result does not hold for p = 1. As an interesting historical note, this density result was fistpublished in a 1964 paper by Meyers and Serrin having the title H = W .

We additionally denote by W p,k

0 (⌦) the closure of C10 (⌦) in the W k

pnorm. Here C1

0 (⌦) is theset of infinitely di↵erentiable functions having compact support in ⌦. Note that in the context ofsecond-order elliptic boundary value problems, we generally do not consider spaces W p,k

0 for k > 1.This is roughly because we only wish to enforce boundary conditions directly on u itself, whileW p,k

0 (⌦) (k � 2) enforces 0 boundary conditions on all derivates of order k � 1 and less. Thesolution to ��u = f , u = 0 on @⌦ thus may lie in H1

0 (⌦) \H2(⌦), but not generally in H20 (⌦).

We also define fractional-order Sobolev spaces. Let 1 p < 1, and for nonnegative real numbers let s = s+ ✓, where s is an integer and 0 < ✓ < 1. We then define the seminorm

|u|pWp,s(⌦) =

X

|↵|=s

ZZ

⌦⇥⌦

|D↵u(x)�D↵u(y)|p

|x� y|n+✓pdx dy (1.6)

and the norm

kukWp,s(⌦) =⇣kukp

W sp (⌦) + |u|p

Wp,s(⌦)

⌘1/p; (1.7)

cf. [37], Definition 1.3.2.1. When p = 2 we set Hs(⌦) = W s

2 (⌦), as usual.

1.3.3 Sobolev imbeddings

It is natural (and often of interest in applications) to ask whether a function lying in given Sobolevspace is also continuous, or lies in some other Sobolev space which in general has stronger integrabilityindex p but weaker di↵erentiability index k. The concept of Sobolev imbeddings is intended to answer


such questions. Our intention is to give a brief introduction to the idea of such imbeddings; see [1, 2]as above for more details.

We first give a standard example. The function u : R2 ! R defined by u(x) = log log |x| canbe shown to lie in H1(⌦) for any finite domain ⌦ (including ones containing the origin), but it isnot continuous at the origin. Thus H1 functions need not be continuous. This is important becausethe weak form of Poisson’s problem, and second-order elliptic problems in general, is solved overH1(⌦) or H1

0 (⌦). We thus might naturally ask what conditions are necessary to ensure continuoussolutions of Poisson’s problem.

Given a bounded domain ⌦, we let Ck

B(⌦) denote those functions which are in Ck(⌦) with

uniformly bounded derivatives of all orders up to k. Ck

B(⌦) is a Banach space with norm kuk

CkB(⌦) =

max|↵|k kD↵ukL1(⌦). Note that Ck

B(⌦) ⇢ W k

1(⌦), but not vice-versa. We write CB(⌦) = C0B(⌦).

Given Banach spaces X and Y , we say that X is imbedded into Y and write X ! Y if X is avector subspace of Y and kukY . kukX .

Theorem 1.3.1 (Sobolev Imbedding Theorem [2]). Suppose that ⌦ satisfies the cone condition.Then

1. If either kp > d or k = d and p = 1, then

W k,p(⌦) ! CB(⌦). (1.8)

If in addition 1 k d, then

W k,p(⌦) ! Lq(⌦), p q 1. (1.9)

2. If kp = d, then

W k,p(⌦) ! Lq(⌦), p q < 1. (1.10)

3. If kp < d and either d� kp d or p = 1, then

W k,p(⌦) ! Lq(⌦), p q p⇤ =dp

d� kp. (1.11)

Here the imbedding constants generally depend on the involved parameters and may degenerate (blowup) as limit cases are approached.

We give a couple of examples. Consider (1.8) with k = 1. We may conclude that if u 2 W 1,p(⌦)with p > d, then u is continuous with kukL1(⌦) . kukW 1,p(⌦). In particular, in two space dimensionsu is continuous if u 2 W 1,p(⌦) for any p > 2, while when d = 3 u is continuous if u 2 W 1,p(⌦) forsome p > 3. We next take k = 2, that is, we ask what integrability index the second derivatives ofu must possess in order for u to be continuous. For d = 2, we have d = k = 2 and thus (1.8) holdswith p = 1. For d = 3 we require 2p > 3, or p > 3/2 in order to ensure that u is continuous.

It can be important to note that the constants involved in the Sobolev imbeddings can blow upnear limit cases, though often in a known fashion. To illustrate this, we follow the discussion ofLemma 5.4 of [48]. As an example, if d = 2 and u 2 H1

0 (⌦) and 2 p < 1, then

kukLp(⌦) .ppkrukLd(⌦).


Note that the constantpp blows up as p " 1. Combining the above estimate for correctly chosen p

with finite element inverse estimates yields a discrete Sobolev inequality when d = 2. In particular,assume that ⌦ is polyhedral with Th a quasi-uniform mesh and Vh ⇢ H1

0 (⌦) an associated Lagrangefinite element space as above. Then for � 2 Vh,

k�kL1(⌦) . h�2/pk�kLp(⌦), � 2 Sh.

We then let p = ln 1h. In this case h�2/p = (e� lnh)2/ lnh . 1, so

k�kL1(⌦) . (ln 1/h)1/2kr�kL2(⌦).

The above inequality does not hold for general functions in H1, as we have seen above. However,finite element functions (at least those we consider here) are continuous, so we can get away withan “almost”-Sobolev inequality. We pay for it with the factor ln 1

h, which blows up as h ! 0 (albeit

slowly). This discrete Sobolev inequality is well-known and fairly widely used in the finite elementliterature, though it is mainly useful for analyzing FEM for 2D problems. (A 3D version holds butrequires measuring the gradient in L3 instead of L2, which is not nearly as interesting because H1

is the natural energy space and W 1,3 generally isn’t!)

1.3.4 Basic regularity result for elliptic PDE’s

We conclude by giving a more precise statement of the shift theorem (1.1); cf. Chapter 6 of thetextbook [30] of Evans. The monograph [34] by Gilbarg and Trudinger is also a good resource forbasic material about elliptic PDE.

Consider the divergence-form elliptic operator

Lu := �dX

i,j=1

(aij(x)uxi)xj +dX

i=1

bi(x)uxi + c(x)u.

Assume first that the coe�cients aij , bi, and c are all in L1(⌦), and that the bilinear form

a(u, v) :=

Z

⌦(

dX

i,j=1

aij(x)uxivxj + vdX

i=1

bi(x)uxi + c(x)uv) dx

is coercive and continuous over H10 (⌦) in the sense that

kuk2H1(⌦) . a(u, u), a(u, v) . kukH1(⌦)kvkH1(⌦), u, v 2 H1

0 (⌦).

We also assume that the operator is (strongly) elliptic, that is, that the matrix A = [aij ] of coe�cientsis uniformly bounded and positive definite in the sense that there are constants �,⇤ > 0 such that

dX

i,j=1

|aij(x)| ⇤2, aij(x)⇠i⇠j > �|⇠|2, x 2 ⌦, ⇠ 2 Rd.

The weak form of the PDE Lu = f , u = 0 on @⌦ is: Find u 2 H10 (⌦) such that

a(u, v) = (f, v), v 2 H1(⌦).


Then the Lax-Milgram theorem guarantees existence and uniqueness of u 2 H10 (⌦) satisfying the

above for any linear functional f 2 H�1(⌦) := {f : H10 (⌦) ! R : f(v) . kvkH1

0 (⌦), v 2 H10 (⌦)}

and kukH10 (⌦) . kfkH�1(⌦). Such a u is the weak solution to the strong-form problem Lu = f in ⌦,

u = 0 on @⌦.

Theorem 1.3.2 Assume that aij 2 C1(⌦), bi, c 2 L1(⌦), 1 i, j d, that f 2 L2(⌦), thatu 2 H1

0 (⌦) is the unique weak solution of Lu = f , u = 0 on @⌦ as above, and that @⌦ is C2. Thenu 2 H2(⌦), and

kukH2(⌦) . kfkL2(⌦)

with the constant hidden in . depending only on ⌦ and the coe�cients of L.

The proof is rather long and technical, and outside of the scope of this course. We do howevermake a couple of notes:

1. The basic approach of [30] to regularity proofs is to establish estimates for di↵erence quotientsof ru and take a limit as the mesh size approaches 0.

2. Establishing interior regularity results (that is, proving that u 2 H2(⌦0) for domains ⌦0 ⇢⇢ ⌦)is pretty straightforward, if technical, under the assumptions in Theorem 1.3.2.

3. To establish that u 2 H2(⌦) for domains close to the boundary, the general strategy is toflatten the boundary locally. That is, one first proves regularity if ⌦ has a flat boundary. Onethen maps portions of @⌦ that aren’t flat to a domain having flat boundary and then proceedsas before–with the added di�culty of using the chain rule to take the mapping into account.This is the point where regularity of the boundary comes into play in classical proofs.

We finally state a version of Theorem 1.3.2 assuming higher boundary and coe�cient regularity.

Corollary 1.3.3 In addition to the assumptions of Theorem 1.3.2, assume that aij , bi, c 2 Ck+1(⌦),f 2 Hm(⌦), and @⌦ is Cm+2. Then u 2 Hm+2(⌦) and

kukHm+2(⌦) . kfkHm(⌦),

the constant hidden in . only depending on ⌦, m, and the coe�cients of L.

We also briefly mention other types of estimates for elliptic boundary value problems; cf. [34,Chapters 8 and 9].

1. W p,2 estimates: Assume that aij 2 C1(⌦), bi, c 2 L1(⌦) as in Theorem 1.3.2, that @⌦ isC1,1, and that f 2 Lp(⌦) for some 1 < p < 1. Then the problem (written in strong form) offinding u such that Lu = f , u = 0 on @⌦ has a unique weak solution u 2 W p,2(⌦) \W p,1

0 (⌦)satisfying

kukWp,2(⌦) CpkfkLp(⌦).

Note that the above estimate does not hold for p = 1,1, and that Cp blows up as p " 1or p # 1. (The constant Cp comes from the Marcinkiewicz interpolation theorem at least forsu�ciently smooth domains. I have seen assertions that it behaves as Cp for large p and C

p�1

for p close to 1, but I couldn’t immediately find a reference.)


2. Maximum principles: The weak maximum principle states that if u 2 W 2,1(⌦) satisfies Lu 0(Lu � 0) in ⌦, then

sup⌦

u sup@⌦

u+ (inf⌦

u � inf@⌦

u�).

The rough intuition of the maximum principle is that Lu 0 implies that u is convex, andLu � u that u is concave. This corresponds to our usual intuition from calculus, since Lu 0roughly means that the second derivative of u is positive.

3. Holder estimates for the first derivatives of u: Following [34, Section 8.11], we now assume

that Lu = �u and that �u = g +P

d

i=1 Dxifi. If u is a weak solution with u 2 C1,↵(⌦) and

u|@⌦ = 0, thenkukC1,↵(⌦) . kukC0(⌦) + kgkC0(⌦) + kfkC0,↵(⌦).

Notes for Math 663 Spring 2016demlow/Courses/16_sp/663_sp16/...Notes for Math 663 Spring 2016 Alan...

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